Analytic and numeric implementation of conformal welding

Previously we discussed conformal welding in a fairly general setting. I stated that it had a use in calculating probability distributions of smeared energy densities, my aim is for this post to set the last of the mathematical foundation before I can then delve into the consequences of this. As with last time I will still be primarily following the work of Sharon and Mumford’s paper “2D-Shape Analysis Using Conformal Mapping” due to their excellent discussion on the method of welding for use in calculation, I reiterate though, my blog will be far less thorough than their paper. I’ll just be taking what I will need for subsequent posts.

I described in my last post how one aims to find the map given by

\displaystyle \varphi=\vartheta^{-1}_{+}\circ\vartheta_{-}, \ \ \ \ \ (1)

which is defined on the unit circle. I will note that {\varphi} is a periodic real valued function with positive derivative everywhere. One can prove that these weldings exist using an existence theorem of B. Bojarski and L. Bers which I will give a very vague sketch of to just get the point across.

For any {c<1} and any complex valued function {\mu(z)} which has modulus less than or equal to {c}, the partial differential equation

\displaystyle F_{\bar{z}}=\mu(z)F_{z}, \ \ \ \ \ (2)

has a complex valued solution. Note, the subscripts denote normalised addition and subtraction respectively of the function’s real and imaginary parts. To obtain the function {\mu(z)} from our welding {\varphi} we define an auxiliary function {A} which maps from the inside of our unit disk to itself in the following way

\displaystyle A(re^{i\theta})=re^{i\varphi(\theta)}. \ \ \ \ \ (3)

Then let {\mu(z)=A_{\bar{z}}/A_{z}} which is on {\Delta_{-}}. We use this value of {\mu} to solve our Beltrami equation observing that {\mu} only has support on the unit disk. We find that {F} must be a conformal map on the complement of our unit disk (meaning it extends to {\infty}. This means we can normalise it to have a positive real derivative there. Now, let {\vartheta_{+}} be {F} on {\Delta_{+}}. Recall that {A} satisfies our Beltrami equation on the unit disk so any other solution is {A} followed by an analytic function. To finish we simply let {\varphi_{-}=F\circ A^{-1}} on the unit disk and then this composed with the function A gives {\varphi^{+}} as required. Existence aside, its time for me to go back to butchering results for my own purposes…

In Sharon & Mumford’s paper they specify two methods of welding, I shall give an overview of the second. This algorithm uses a convolution with a singular kernel {1/u} on the real line, multiplies the fourier transformed function {i\epsilon(n)} where {\epsilon(n)} is the negative of the sign of n. One may recognise this as a Hilbert transform, in this case we have a convolution with the cotangent of half the angle.

If we have a square integrable function {f} on the circle we define {H(f)} as the Hilbert transform as described above. We now specify that this function {f} is just the boundary function on a circle of radius {1} and that either inside or outside of this we have that {f_{-}} and {f_{+}} are both complex analytic in their respective regions. Now let {F(\theta)=f_{+}(e^{i\theta})} and realise that {f_{-}(e^{i\theta})=F(\varphi(\theta))} and we can find (after some manipulation) that if we define the operator {K} as follows

\displaystyle K(F)=\frac{i}{2}\left(H(F)-H(F\circ\varphi)\circ\varphi^{-1}\right), \ \ \ \ \ (4)

we get the integral equation {K(F)+F=e^{i\theta}} which can be solved for {F}. One may notice that this will result in a difference in cotangents of differences of angles, there is then potential cause for concern at this coincidence. Fear not though, {K} is a smooth integral operator. While it may be analytically a smooth integral operator, Numerically one can imagine the cotangent terms being problematic, one solution to solve this is to calculate {K} at coincidence, naively one can just perturb from exact coincidence by some {\delta} and it is a simple exercise to show that any infinite terms vanish. By the Fredholm alternative, {F} can be solved for if one inverts the {(I+K)} term in the above equation. The welding problem then becomes the problem of solving the above integral equation.

To go into a little more detail of numerical calculation one can transform this integral equation to a discrete problem in linear algebra by generating a matrix from {K}, adding the identity matrix and then all one has to do is multiply the inverted matrix of those two terms against a vector of {e^{i\theta}}. As described above, a perturbation method of calculating the coincidence terms can give an explicit (and simple) result for the values at coincidence. It may be frustrating for you that I haven’t given a specific formula for {K} but I simply want to get these ideas across, and besides I can’t imagine you are going to leave this post and immediately need to begin calculating conformal weldings…

I must admit I have a real soft spot for the results in the last few posts, the mathematics can be very pretty. However, I think we now have the necessary basis to proceed onto the physics of these problems. There are some really cool results due to the work done in the last three posts, so I will be going onto another tangent in the next post to give a background to the physics then finally we will be describing how the welding relates to these probability distributions and I will finally stop baiting you with these results.

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